I really agree with you. Some companies are already doing it. Pratt & Whitney and Bombardier have air clubs. However, there is a whole other stage.
Last year, the Air Canada Pilots Association and COPA developed a career guide and established pilot scholarships to encourage people to enter the field. But that was not really sponsorship in the strict sense.
Companies would do well to have sponsorships, exactly as you say. It is perfectly possible. The costs would be minimal, but there needs to be a plan. COPA would be ready to work with people. To start programs like that, we could use airports and aerodromes located away from the problem areas we were talking about earlier.